
and at each point of \(\varpi ^{-1}(D)\) , summing in \(\)  the various \(_{\le \varphi }\) 's for the \(\varphi \) 's which coincide near this point.
[Proof of Proposition REF ]
We fix \(x\in D^\)  and denote by \(D_1^\circ \)  the stratum of \(x\)  and by \(\gamma :(,0)\rightarrow (X,x)\)  a curve transversal to \(D_1^\circ \)  at \(x\) .
We first note that it is enough to prove the following statement.
A germ \(\)  at \(x_o\)  which satisfies \(_{D,x_o}\simeq ^_{D,x_o}\)  is uniquely determined by its restriction \(_{D|D_1^\circ }\) .
Indeed, if Assertion REF  is proved, assume \(\gamma ^+(,_{D})\simeq \gamma ^+(^{\prime },^{\prime }_{D})\) . By Corollary REF , we obtain that \((,_{D})_{|D_1^\circ }\simeq (^{\prime },^{\prime }_{D})_{|D_1^\circ }\) . According to the assertion, we then have \(\simeq ^{\prime }\) . Let us check that in such a case, \(_{D}\)  is also uniquely determined by its restriction to \(D_1^\circ \) . Since it can be represented by a matrix with entries in \(_{D}\) , it is enough to prove the assertion for sections \(f\)  of \(_{D}\)  on a neighbourhood of \(x_o\) , which follows then from the description of \(_{D}\)  recalled in the proof of Proposition REF . Indeed, setting \(f=(f_i)_{i=1,\dots ,\ell }\) , the condition \(f_{1|D_1^\circ }=0\)  implies \(f_1=0\) , hence \(f_i=0\)  in \(_{x_o}\)  for each \(i\) , and therefore \(f_i=0\)  in \(_{D_i,x_o}\) .
We will thus prove Assertion REF . Note that, if \(D(\Phi _{x_o})\!\ne \!D_{x_o}\) , Example REF  implies that \(\simeq ^\)  and there is nothing to prove. However, note that the existence of some \(_{D}\)  is essential in this case. We will then assume that \(D(\Phi _{x_o})\!=\!D_{x_o}\) . We start with a particular case.
[Proof of the assertion in a simple case]
We assume here that for all \(\varphi \ne \psi \in \Phi _{x_o}\) , \(\varphi -\psi \)  has poles along each component of \(D_{x_o}\) . Let \((,_)_{\varpi ^{-1}(D_1^\circ )}\)  be the Stokes-filtered local system on \(\varpi ^{-1}(D_1^\circ )\)  corresponding to \(_{|D_1^\circ }\) . By our assumption on \(\Phi _{x_o}\) , it is the pullback by the natural projection
\(\varpi ^{-1}(D_1^\circ )=(S^1)^\ell \times \lbrace 0\rbrace \times (0,\epsilon )^{\ell -1}\times B^{m-\ell }\rightarrow (S^1)^\ell \times \lbrace 0\rbrace ^\ell \times B^{m-\ell }=\varpi ^{-1}(D^{(x_o)})\) 
of a Stokes-filtered local system \((,_)_{\varpi ^{-1}(D^{(x_o)})}\)  on \(\varpi ^{-1}(D^{(x_o)})\) , because in each summation procedure recalled in (REF ) above, there is only one term. As a consequence, \((,_)_{\varpi ^{-1}(D^{(x_o)})}\)  is uniquely determined from \((,_)_{\varpi ^{-1}(D_1^\circ )}\) . We conclude that \(\)  is uniquely determined by its restriction to \(D_1^\circ \) .
Before giving the proof in general, let us recall the level structure of Stokes-filtered local systems indexed by \(\Phi _{x_o}\)  on \(\partial X\)  ([1]}, [2]}). It is parallel to that for \(D\) -meromorphic flat bundles considered in Proposition REF . In order to simplify the notation, we will assume that \(0\in \Phi _{x_o}\) , that we take as base point \(\varphi _o\) , and we use the notation as in (REF ). Let us start with a Stokes-filtered local system \((,_)\)  indexed by \(\Phi _{x_o}\)  on \(\varpi ^{-1}(D^{(x_o)})\) . It induces a filtration \(_\preceq \)  indexed by \(C=C(\varphi _o)\subset \) , where the order \(c\preceq c^{\prime }\)  on the latter set at a point of \(\varpi ^{-1}(D^{(x_o)})\)  is defined as the property that \(\exp ((c-c^{\prime })x^{-_o})\)  has moderate growth near this point. By definition, we have
\(_{\preceq c}=\sum _{{\psi \in \Phi _{x_o}\\ c(\psi )\le c}}_{\le \psi }.\) 
Moreover, for every \(c\in C\)  (recall that we can assume \(\#C\ge 2\) , Lemma REF ), the graded sheaf \(_c\)  is a locally constant sheaf, on which the filtration \(_\)  induces a Stokes-filtration whose jumps are contained in \(\Phi _{x_o}(c)\) . Lastly, for \(\psi \in \Phi _{x_o}(c)\) , \(_{\le \psi }\)  is the pullback of \((_c)_{\le \psi }\)  by the projection \(_{\preceq c}\rightarrow _c\) . By the Riemann-Hilbert correspondence, \((_c,(_c)_)\)  corresponds to \(_c\)  considered in Proposition REF .
A similar structure is obtained for a Stokes-filtered local system on \(\varpi ^{-1}(D_1^\circ )\) , by taking pullback and summation of \((,_)_{\varpi ^{-1}(D_1^\circ )}\)  as above.
Proof of Assertion REF  in the general case
We argue by induction on the rank \(r\)  of \(\) . It remains to consider the case where \(D(\Phi _{x_o})=D_{x_o}\) . Given \(,^{\prime }\)  with \(_{D}\simeq ^{\prime }_{D}\simeq ^_{D}\) , assume that \(_{|D_1^\circ }\simeq ^{\prime }_{|D_1^\circ }\) . It follows that the Stokes-filtered local systems \((,_)_{\varpi ^{-1}(D_1^\circ )},(^{\prime },^{\prime }_)_{\varpi ^{-1}(D_1^\circ )}\)  are isomorphic, hence we have an isomorphism on \(\varpi ^{-1}(D_1^\circ )\) :
\(\bigl (,_\preceq ,(_c,(_c)_)_{c\in C}\bigr )_{\varpi ^{-1}(D_1^\circ )}\overset{\textstyle (*)}{\simeq }\bigl (^{\prime },^{\prime }_\preceq ,(_c^{\prime },(_c^{\prime })_)_{c\in C}\bigr )_{\varpi ^{-1}(D_1^\circ )}.\) 
Arguing as in the special case for \(_\preceq \)  instead of \(_\le \) , we obtain an isomorphism \((,_\preceq )_{\varpi ^{-1}(D^{(x_o)})}\simeq (^{\prime },^{\prime }_\preceq )_{\varpi ^{-1}(D^{(x_o)})}\)  and, for each \(c\in C\) , the induced isomorphism \(_c_{\varpi ^{-1}(D^{(x_o)})}\simeq _c^{\prime }_{\varpi ^{-1}(D^{(x_o)})}\)  corresponds to that induced by \((*)\)  (by pullback by the projection). Since \(_{c,D}\simeq ^{\prime }_{c,D}\simeq ^_{c,D}\)  have rank \(<r\) , we conclude by induction that the component
\((*)_c:(_c,(_c)_)_{\varpi ^{-1}(D_1^\circ )}(_c^{\prime },(_c^{\prime })_)_{\varpi ^{-1}(D_1^\circ )}\) 
of \((*)\)  comes from an isomorphism \(_c\simeq ^{\prime }_c\) , that is, an isomorphism on \(\varpi ^{-1}(D^{(x_o)})\) :
\((_c,(_c)_)_{\varpi ^{-1}(D^{(x_o)})}\simeq (_c^{\prime },(_c^{\prime })_)_{\varpi ^{-1}(D^{(x_o)})}.\) 
As a consequence, the isomorphism \(_{\preceq c}^{\prime }_{\preceq c}\)  sends \(_{\le \psi }\)  isomorphically to \(^{\prime }_{\le \psi }\)  on \(\varpi ^{-1}(D^{(x_o)})\)  for any \(\psi \in \Phi _{x_o}(c)\) : indeed, we have a commutative diagram
\(@R=2mm{_{\preceq c}[r]^-\sim [dd]&^{\prime }_{\preceq c}[dd]\\&\\_c[r]^-\sim &_c^{\prime }\\(_c)_{\le \psi }[r]^-\sim @{}[u]|\cup &(_c^{\prime })_{\le \psi }@{}[u]|\cup }\) 
and \(_{\le \psi }\)  \(^{\prime }_{\le \psi }\)  is the pullback of \((_c)_{\le \psi }\)  \((_c^{\prime })_{\le \psi }\)  by the projections \(_{\preceq c}\rightarrow _c\)  and \(^{\prime }_{\preceq c}\rightarrow _c^{\prime }\) .
We conclude that \((,_)\simeq (^{\prime },^{\prime }_)\)  on \(\varpi ^{-1}(D^{(x_o)})\) , as was to be proved.

Global results
We now go back to the global setting of Section REF . In particular, \(D\)  is connected and \(^\)  is fixed. The following result is due to J.-B. Teyssier, who has obtained a stronger form, only in dimension two however.
[J.-B. Teyssier [3]}]
Given \(^\) -marked \(D\) -meromorphic flat bundles \((,_{D})\)  and \((^{\prime },^{\prime }_{D})\)  on \(X\) , we have \((,_{D})\simeq (^{\prime },^{\prime }_{D})\)  if and only if there exists a point \(x\)  in \(D^\)  and a germ of curve \(\gamma :(,0)\rightarrow (X,x)\)  transverse to \(D\)  at \(x\)  such that \((\gamma ^+,_{0})\simeq (\gamma ^+^{\prime },^{\prime }_{0})\) .
For any two germs \((,_{D}),(^{\prime },^{\prime }_{D})\) , let \(D^{\prime }\subset D\)  be the subset of points where they are isomorphic. Assume that it is not empty. It is open by definition. Let us show that it is closed. Let \(x_o\in {D^{\prime }}\) .

Either \(x_o\in D^\) , then \(D^{\prime }\)  contains a point \(x^{\prime }\)  in the component of \(D^\)  containing \(x_o\) , hence also contains this whole component according to Corollary REF , and therefore \(D^{\prime }\ni x_o\) ,

or \(x_o\notin D^\) , and we apply Proposition REF  to also conclude that \(x_o\in D^{\prime }\) .

Therefore, \(D^{\prime }\)  is closed, hence equal to \(D\) . The “only if” part of the proposition is clear, and for the “if” part we apply the previous argument, since we know by Corollary REF  that \(D^{\prime }\ne \emptyset \) .
We will make use of the following consequence, which is a weaker version of [3]}. Although its statement is local, it relies on the global result of Corollary REF .
[J.-B. Teyssier]
Assume that \(D\)  is smooth. Let \(\Psi _{x_o}\subset _{X,x_o}(*D)/_{X,x_o}\)  be a (not necessarily good) finite subset, and, for any \(\psi \in \Psi _{x_o}\) , let \(_\psi \)  be a germ of regular \(D\) -meromorphic flat bundle at \(x_o\) . Set \(=\bigoplus _{\psi \in \Psi _{x_o}}(^\psi \otimes _\psi )\) . Let \((,_{D}),(^{\prime },^{\prime }_{D})\)  be two \(\) -marked \(D\) -meromorphic flat bundles in the neighbourhood of \(x_o\)  and let \(\gamma :(,0)\rightarrow (X,D)\)  be a germ of curve whose image is not contained in \(D\) . Then
\((,_{D})\simeq (^{\prime },^{\prime }_{D})\iff \gamma ^+(,_{D})\simeq \gamma ^+(^{\prime },^{\prime }_{D}).\) 
The point is to prove the implication \(\Leftarrow \)  and we assume that the isomorphism of the left-hand side holds for some curve \(\gamma \) . We shall denote by \(X\)  a small (connected) neighbourhood of \(x_o\)  where all data are defined. Let us first assume that \(\gamma (0)=x_o\) . There exists ([2]}) a projective modification \(:X^{\prime }\rightarrow X\)  which is an isomorphism away from \(D\)  and the normal crossing divisor \(D^{\prime }:=^{-1}(D)\)  such that \(^+\)  is good. This result is much easier than the general result on the resolution of turning points obtained by K. Kedlaya [6]} (and T. Mochizuki [1]} in the algebraic case). We can moreover assume that \(\gamma \)  lifts as a curve \(\gamma ^{\prime }\)  which is transverse to \(^{-1}(x_o)\)  at a smooth point. We can apply Corollary REF  to \(^+(,_{D}),^+(^{\prime },^{\prime }_{D})\) , and deduce that both are isomorphic. It follows from Lemma REF  that \((,_{D}),(^{\prime },^{\prime }_{D})\)  are isomorphic.
Assume now that \(x:=\gamma (0)\ne x_o\) . The first part of the proof applied at \(x\)  shows that \((,_{D})_{|(x)}\simeq (^{\prime },^{\prime }_{D})_{|(x)}\) . Let \(U\)  be the open subset of \(D\)  on which \(\)  is good. This is the complement in \(D\)  of the union of the zero sets of the meromorphic functions \(\psi -\eta \)  for \(\psi ,\eta \in \Psi _{x_o}\)  and \(\psi \ne \eta \) . Since \(D\)  is smooth, this set is connected. By Corollary REF , we deduce an isomorphism \((,_{D})_{|U}\simeq (^{\prime },^{\prime }_{D})_{|U}\) . We conclude with Lemma REF .

Proof of the main results
We consider the setting and notation of Theorem .
The most degenerate case
If all coordinates of \(t_o\)  coincide, there is only one class \((\gamma _{t_o}^+,\gamma _{t_o}^+_{T})\)  since, up to a twist by \(^{t_{o,1}/z}\) , \(\gamma _{t_o}^+\)  has a regular singularity at \(z=0\) . The restriction induces then a surjective map between the two corresponding sets of isomorphism classes, and the injectivity is a consequence of the following more precise proposition.
Assume that the connected open subset \(U\)  contains such a \(t_o\) . Let \((_U,_{U})\in \Gamma (U,_T())\) . Then \(_{U}\)  can be lifted (in a unique way) as an isomorphism \(_U:_U_{|U}\) .
We will argue when \(t_o\)  is the origin of \(T\) , the other cases being obtained by an exponential twist. Let \(:X^{\prime }\rightarrow X\)  be the blowing-up of the origin in \(X\)  and let \(T^{\prime }\)  be the strict transform of \(T\)  by \(\) , so that \(_{|T^{\prime }}:T^{\prime }\rightarrow T\)  is nothing but the blowing-up of \(T\)  at the origin. Let \(E=^{-1}(0)\simeq ^n\)  be the exceptional divisor, so that \(D:=E\cup T^{\prime }\)  is a divisor with normal crossings.
There is a chart with coordinates \((u_1,\dots ,u_n,\zeta )\)  such that \(E\)  is defined as \(\zeta =0\)  and \(\)  is given by \((u_1,\dots ,u_n,\zeta )(u_1\zeta ,\dots ,u_n\zeta ,\zeta )\) , so that \(^+\)  has regular singularities along \(E\)  away from \(E\cap T^{\prime }\simeq ^{n-1}\) . On the other hand, one checks that \(^+\)  is good on a Zariski dense open subset of \(E\cap T^{\prime }\) , by computing in a chart with coordinates \((v_1,\dots ,v_n,\zeta )\)  where \(\)  is given (say) by \((v_1,\dots ,v_n,\zeta )(v_1,v_1v_2,\dots ,v_1v_n,v_1\zeta )\) , the meromorphic functions \(^*((t_i-t_j)/z)\) .
Let \(t^{\prime }_o\)  be a point in this open subset of \(E\cap T^{\prime }\) , that we can also consider as a point in \(X^{\prime }\) . At this point, \(^+\)  has thus regular singularities along one of the components of \(D\)  passing through \(t^{\prime }_o\) . From Example REF  we conclude that \(^+_U\simeq ^+\)  in some open neighbourhood of \(t^{\prime }_o\) . Intersecting this neighbourhood with \(U\Delta \)  gives a non-empty open subset in \(U\Delta \)  where \(_{U}\)  can be lifted as an isomorphism \(_U\) . We conclude with Corollary REF , applied to the connected open subset \(U\Delta \) , that \(_{U}\)  can be lifted as an isomorphism \(_{U\Delta }:_{|U\Delta }_{|U\Delta }\) . We now conclude with Lemma REF .

Proof of a variant of Theorem  in a special case
As an example for the method of proof of Theorem , we develop in this section a low-dimensional case, obtained by a generic two-dimensional slice of the pair \((T,\Delta )\)  considered in the general case.
The setting is as follows. We have \(\dim T=2\) , with coordinates \(t=(t_1,t_2)\) . We fix \(a,b\in \)  such that \(a,b,a-b\ne 0\)  and we consider the following functions:
\(f_1(t)=t_1,\quad f_2(t)=t_1+t_2,\quad f_3(t)=t_1+a,\quad f_4(t)=t_2+b,\) 
and the elementary \(T\) -meromorphic flat bundle
\(=\bigoplus _{i=1}^4(^{-f_i(t)/z}\otimes _i),\) 
with \(_i\)  regular along \(T=\lbrace z=0\rbrace \) . Then \(\)  is good away from \(\Delta =\lbrace t_2=0\rbrace \) , so that the stratum is reduced to \(\Delta \)  and we set \(U=T\) . We also set
\(_o=\bigl (^{-f_1(t)/z}\otimes (_1\oplus _2)\bigr )\oplus \bigoplus _{i=3}^4(^{-f_i(t)/z}\otimes _i),\) 
so that \(_o\)  is a good decomposed \(T\) -meromorphic flat bundle.
We denote by \(:X_1\rightarrow X\)  the blow-up of the origin in \(X\) . We set \(E_1\!=\!^{-1}(0)\!\simeq \!^2\) , we denote by \(T_1\)  the strict transform of \(T\)  by \(\) , so that \(_{1|T_1}:T_1\rightarrow T\)  is the blow-up of the origin in \(T\) . The strict transform \(\Delta _1\)  of \(\Delta \)  is contained in \(T_1\)  and intersects \(_{1|T_1}^{-1}(0)=E_1\cap T_1\simeq ^1\)  at one point \(\delta _1\) . The restriction \(_{1|\Delta _1}:\Delta _1\rightarrow \Delta \)  is an isomorphism.
<FIGURE>We denote by \(:X_2\rightarrow X_1\)  the blowing-up of \(\Delta _1\)  in \(X_1\) . We set
\(E_2=^{-1}(\Delta _1)\simeq ^1\times \Delta _1,\) 
we denote by \(E_{1,2}\)  the strict transform of \(E_1\) , so that \(_{2|E_{1,2}}:E_{1,2}\rightarrow E_1\)  is the blow-up of \(\delta _1\)  in \(E_1\)  and \(_{2|E_{1,2}}^{-1}(\delta _1)\simeq ^1\times \lbrace \delta _1\rbrace \) . It is standard to check that \(E_{1,2}\) , being the complex blowing up of \(^2\)  at one point, is simply connected.One can regard \(E_{1,2}\)  as the union of \(^2\)  and the union of two \(^1\)  which intersect at one point; both sets are simply connected, and the second one has an open neighbourhood \(W\)  which retracts onto it, hence is also simply connected; since \(W\cap ^2\)  is connected, van Kampen theorem gives the result. Let \(T_2\)  be the strict transform of \(T_1\) . Then \(_{2|T_2}:T_2\rightarrow T_1\)  is an isomorphism since it is the blow-up of \(\Delta _1\)  of codimension one in \(T_1\) . We also have \(T_2\cap E_{1,2}\simeq ^1\)  since it is the blow-up of \(^1\)  at \(\delta _1\) . We thus regard \(\delta _1\)  as a point in \(X_2\)  and \(\Delta _1\)  as the subset \(T_2\cap E_2\)  in \(X_2\) . The geometric setting at \(\delta _1\)  in \(X_2\)  is pictured in Figure REF .
We set \(=\circ :X_2\rightarrow X\) . We denote by \(\varpi :X_2\rightarrow X_2\)  the real oriented blowing up of \(X_2\)  along the components \((T_2,E_{1,2},E_2)\)  of \(D:=^{-1}(T)\) .
The sheaf \(_D(^+_o)\)  is locally constant on \(^{-1}(T)\) .
Using the notation of Section REF , we note that the pushforward \(_*\)  induces an isomorphism \(_D(^+_o)^{-1}_T(_o)\) . By Proposition REF , \(_T(_o)\)  is a locally constant sheaf on \(T\) , hence so is \(_D(^+_o)\)  on \(D\) . The assertion follows then from Theorem REF .
The \(D\) -meromorphic flat bundle \(^+\)  is good and the sheaf \(_D(^+)\)  is constant when restricted to \(E_{1,2}\) .
Since \(E_{1,2}\)  is simply connected, it is enough to prove local constancy. Let us write down the charts of the various blow-ups. We cover \(X_1\)  by three charts \(X_1(i)\)  (\(i=1,2,3\) ), with coordinates \((u_i,v_i,\zeta _i)\)  so that \(_1\)  is given respectively by the formulas:
\(X_1(1):{\left\lbrace \begin{array}{ll}t_1=u_1\zeta _1,\\t_2=v_1\zeta _1,\\z=\zeta _1,\\\end{array}\right.}\quad X_1(2):{\left\lbrace \begin{array}{ll}t_1=u_2v_2,\\t_2=v_2,\\z=v_2\zeta _2,\\\end{array}\right.}\quad X_1(3):{\left\lbrace \begin{array}{ll}t_1=u_3,\\t_2=u_3v_3,\\z=u_3\zeta _3.\\\end{array}\right.}\) 
We note that the charts \(X_1(1)\)  and \(X_1(2)\)  can be regarded as contained in \(X_2\) , since they do not intersect the center \(\Delta _1=\lbrace u_3=\zeta _3=0\rbrace \)  of the blowing-up \(_2\) .


In the chart \(X_1(1)\) , we have \(X_1(1)\cap T_1=\emptyset \)  and \(X_1(1)\cap E_1=\lbrace \zeta _1=0\rbrace \) . One checks that \(_1^+\)  is good there, and more precisely \(\exp _1^*(-t_2/z)\)  is holomorphic there and the block-diagonal morphism \((,\exp _1^*(-t_2/z),,)\)  induces an isomorphism between \(_1^+_o\)  and \(_1^+\) .


In the chart \(X_1(2)\) , we have \(X_1(2)\cap T_1=\lbrace \zeta _2=0\rbrace \)  and \(X_1(2)\cap E_1=\lbrace v_2=0\rbrace \) . One checks that \(_1^+\)  is good there. Although \(\exp _1^*(-t_2/z)=\exp (-1/\zeta _2)\)  is not holomorphic near \(E_1\cap T_1\) , we claim that the block-diagonal morphism corresponding to \((,\exp _1^*(-t_2/z),,)\)  induces an isomorphism between \(^(_1^+_o)\)  and \(^(_1^+)\)  on \(\varpi ^{-1}(E_1\cap T_1)\) . Indeed, this morphism only affects the blocks \(ij\)  with \(i\ne j\)  and \(i\)  or \(j\)  equal to 2. The blocks 12 and 21 of a section of \(^(_1^+)\)  are zero, since \(\exp _1^*((f_2-f_1)/z)=\exp _1^*(-t_2/z)\)  is nowhere of rapid decay, and so are the blocks 12 and 21 of \(^(_1^+_o)\) . For the blocks 23, 32, 24 and 34, which may be nonzero, multiplying by \(\exp _1^*(-t_2/z)\)  does not affect the leading term of the exponential, hence does neither affect the rapid decay condition.

At this step, we have proved that \(^(_1^+_o)\)  and \(^(_1^+)\)  are isomorphic on \(\varpi ^{-1}(E_{1,2})\varpi ^{-1}(E_2)\) , hence so are \(_D(^+_o)\)  and \(_D(^+)\)  on \(E_{1,2}E_2\) .


We now blow up the chart \(X_1(3)\)  along the ideal \((v_3,\zeta _3)\)  of \(\Delta _1\) , giving rise to the charts \(X_2(3\textup {a})\)  and \(X_2(3\textup {b})\)  of \(X_2\) , with respective coordinates \((u_3,w_1,\eta _1),(u_3,w_2,\eta _2)\)  satisfying
\(X_2(3\textup {a}):{\left\lbrace \begin{array}{ll}v_3=w_1\eta _1,\\\zeta _3=\eta _1,\end{array}\right.}\quad X_2(3\textup {b}):{\left\lbrace \begin{array}{ll}v_3=w_2,\\\zeta _3=w_2\eta _2.\end{array}\right.}\) 

Then one checks that \(^+\)  is good and the same argument as given in (REF ) above gives an isomorphism on \(\varpi ^{-1}(E_{1,2})\)  between \(^(^+_o)\)  and \(^(^+)\) , hence \(_D(^+_o)\)  and \(_D(^+)\)  are isomorphic on \(E_{1,2}\) . The lemma follows from Lemma REF .
Proof of the surjectivity of \(\gamma _{t_o}^+\)
Recall that, here, \(S_o=\Delta \)  and that we have fixed \(t_o\in \Delta \) .

Step one: extension to \(E_{1,2}\)
The curve \(\gamma _{t_o}\)  lifts as a curve \(\gamma _{t^{\prime }_o}:(,0)\rightarrow (X_2,t^{\prime }_o)\)  transverse to \(E_{1,2}\)  at a point \(t^{\prime }_o\)  in \(D^\) . We have \(_0(i_{t_o}^+)=_0(i_{t^{\prime }_o}^+^+)\) . By Proposition REF , this set is identified with \(i_{t^{\prime }_o}^{-1}_D(^+)\) . Since \(_D(^+)_{|E_{1,2}}\)  is constant, according to Lemma REF  and Theorem REF , this set is also identified with \(\Gamma (E_{1,2},_D(^+))\) . An element of \(_0(i_{t_o}^+)\)  defines thus a pair \((,_{D})\)  on some neighbourhood of \(E_{1,2}\)  in \(X_2\) .

Step two: extension to \(E_{1,2}\cup \Delta _1\)
We apply now Proposition REF  to our section of \(_D(^+)_{|E_{1,2}}\) . Since \(\Delta _1\)  retracts to a neighbourhood of \(\delta _1\) , the section extends to a neighbourhood of \(\Delta _1\)  in a unique way.

Step three: extension to \(E_{1,2}\cup E_2\)
The restriction to \((E_{1,2}\cap E_2)\!\!\lbrace \delta _1\rbrace \)  of the section of \(_D(^+)_{|E_{1,2}}\)  constructed in Step one extends uniquely as a section of \(_D(^+)_{E_2\Delta _1}\)  by Proposition REF . The restriction of the section constructed in Step two to a punctured neighbourhood of \(\Delta _1\)  in \(E_2\)  coincides with the restriction to this punctured neighbourhood of this new extension, by uniqueness, since they coincide in the neighbourhood of \(\delta _1\) .

Step four: extension to \(D=E_{1,2}\cup E_2\cup T_2\)
The section constructed in Step three exists in some neighbourhood of \(E_{1,2}\cup E_2\)  in \(D\) . There exists a fundamental basis of neighbourhoods \(U\)  of \(E_{1,2}\cup E_2\)  in \(D\)  such that the inclusion \(U\subset D\)  induces an isomorphism \(\pi _1(U)\pi _1(D)\) , as seen by taking the pullback by \(\)  of a suitable basis of neighbourhoods of \(\Delta \)  in \(T\) . By arguing like in Corollary REF , the section constructed in Step three, defined on such a neighbourhood \(U\) , extends in a unique way to a section of \(_D(^+)\)  defined on \(D\) . Lemma REF  enables us to conclude.

Proof of Theorem 
We now consider the general case in Theorem . Our first aim is to prove, in the context of Theorem , an analogue of Propositions REF  and REF  on each stratum of \(\Delta \) . Let \(t_o\in \Delta \subset T\)  and let \(S(t_o)\)  be its stratum.
When restricted to \(S(t_o)\) , the sheaf \(_T()\)  is a locally constant sheaf of pointed sets, and its fiber at \(t_o\)  is in bijection with \(_0(\gamma _{t_o}^+)\) .
We fix \(t_o\in \Delta \subset T\)  and we work locally at \(t_o\) . We can decompose \(\lbrace 1,\dots ,n\rbrace \)  as \(\bigsqcup _{r\in R}I_r\)  such that, for every \(r\in R\) , we have \(\lbrace i,j\rbrace \subset I_r\)  if and only if \(t_{o,i}=t_{o,j}\) . For each \(r\in R\) , we choose an element in \(I_r\)  that we denote by \(r\)  and we set \(I^{\prime }_r=I_r\lbrace r\rbrace \) . We then set \(p=\#R=\dim S(t_o)\) , \(m=n-p\) , and
\(_o=\bigoplus _{r\in R}(^{-t_r/z}\otimes _{I_r}),\quad _{I_r}:=\bigoplus _{i\in I_r}_i.\) 
We fix a neighbourhood of \(t_o\)  in \(T\)  of the form \(V\times W\) , such that \(V\)  has the coordinates \((t_r)_{r\in R}\)  and \(W\)  has coordinates \((\tau _i^r)_{r\in R,\,i\in I^{\prime }_r}\) , so that for \(r\in R\)  and \(i\in I^{\prime }_r\) , we have \(t_i=t_r+\tau _i^r\) , and small enough such that \(\Delta \cap (V\times W)\)  is given by the equations \(\tau _i^r=0\)  (\(r\in R,\,i\in I^{\prime }_r\) ) and \(\tau _i^r-\tau _j^r\)  (\(r\in R,\,i\ne j\in I^{\prime }_r\) ). We have \(S(t_o)\cap (V\times W)=V\times \lbrace 0\rbrace \) . We now denote by \(T\)  this neighbourhood, and set \(X=(V\times W)\times _z=:V\times Y\) .
Let \(_1:X_1\rightarrow X\)  be the blowing up of this stratum, that of the ideal
\(\bigl ((\tau _i^r)_{r\in R,\,i\in I^{\prime }_r},z\bigr ).\) 
We have \(X_1=V\times Y_1\)  with obvious notation. Every object below is a product of \(V\)  with the corresponding object in \(Y_1\) . The pullback \(D_1:=_1^{-1}(T)\)  is the union of the exceptional divisor \(E_1=_1^{-1}(S(t_o))\)  (in this local setting, \(E_1\simeq S(t_o)\times ^m\) ), and the strict transform \(T_1\)  of \(T\) . The exceptional divisor of \(_{|T_1}:T_1\rightarrow T\)  is equal to \(E_1\cap T_1\simeq S(t_o)\times ^{m-1}\) . Moreover, \(T_1\)  is a disc-bundle in the normal bundle of \(E_1\cap T_1\)  in \(T_1\) . Similarly, the strict transform \(\Delta _1\subset T_1\)  of \(\Delta \)  is a disc-bundle over \(\delta _1:=E_1\cap \Delta _1\)  (the product of an arrangement of projective hyperplanes in \(^{m-1}\)  with \(S(t_o)\) ). We denote by \(\Delta ^{\prime }_1\)  the union of the two-by-two intersections of the components of \(\Delta _1\)  (it has codimension three in \(X_1\) ) and by \(\delta ^{\prime }_1\)  its intersection with \(E_1\) . Lastly, we set \(X_1^\circ =X_1\Delta ^{\prime }_1\) , and similarly for the other objects. In particular, \(\delta _1^\circ \)  is non-singular and \(\Delta _1^\circ \)  is a disc-bundle over it.
We now argue as for the simple case of Section REF . We denote by \(_2:X_2^\circ \rightarrow X_1^\circ \)  the blowing-up of \(\Delta _1^\circ \)  in \(X_1^\circ \)  and we obtain \(D_2^\circ :=E_2^\circ \cup E_{1,2}^\circ \cup T_2^\circ \) . Then Lemmas REF  and REF  hold in this setting. The isomorphism to be considered is block-diagonal with blocks indexed by \(R\) , each block having the form \((,(\exp (-^*\tau _i^r))_{i\in I^{\prime }_r})\) .
For the surjectivity of \(\gamma _{t_o}^+\) , there is no change to be done in Step one of Section REF . For the other steps, we use the disc-bundle structure over the intersection with \(E_{1,2}^\circ \)  of all the objects involved, instead of the structure of a product with \(\) . The same homotopy argument applies.
Let us now conclude with the surjectivity of \(\gamma _{t_o}^+\) . Given \((^{t_o},_{0})\)  with model \(\gamma _{t_o}^+\) , corresponding to an element of
\(_0(\gamma _{t_o}^+)\simeq _0(\gamma _{t^{\prime }_o}^+^+)\simeq _{D_2^}(^+)_{t^{\prime }_o},\) 
we have extended it in a unique way as a section of \(_{D_2^\circ }(^+)\) , which corresponds thus to a pair \((_2^\circ ,_{{D_2^\circ }})\)  with model \(^+\)  satisfying \(\gamma _{t^{\prime }_o}^+(_2^\circ ,_{{D_2^\circ }})=(^{t_o},_{0})\) . Applying \(_{2+}\)  and according to Lemma REF , we obtain a pair \((_1^\circ ,_{{D_1^\circ }})\)  on \(X_1^\circ \) . Due to the theorem of B. Malgrange (Proposition REF ), this pair extends in a unique way as a pair \((_1,_{{D_1}})\)  on \(X_1\) . Lastly, applying \(_{1+}\)  and according to Lemma REF , we obtain a pair \((,_{T})\)  as wanted.
On the other hand, injectivity of \(\gamma _{t_o}^+\)  is given by the proof above, since there is no choice in any extension procedure. This shows that, on \(V\) , the sheaf \(_T()_{|S(t_o)}\)  is constant, with fibre given by applying \(\gamma _{t_o}^+\) .
[End of the proof of Theorem ]
We first define the notion of a star-shaped open set. Let \(S_o\)  be a stratum of \(\Delta \)  and set \(S_o^\star =\bigcup _{S\supset S_o}S\)  be its star (where \(S\)  varies in the set of strata of the natural stratification of \(\Delta \) , so that \(S\)  is a linear subspace). By choosing the coordinates as in the proof of Proposition REF , we find a product decomposition \(S_o^\star \!\simeq \!S_o\!\times \!^m\) , and we consider the corresponding projection \(p_o\,{:}\,S_o^\star \!\rightarrow \!S_o\) . We endow \(^m\)  with its standard Euclidean metric.
An open subset \(U\subset T\)  is said to be star-shaped with respect to \(U\cap S_o\)  if \(U\)  contains the Euclidean ball centered at the origin of the linear subspace \({S(t)}\cap p_o^{-1}(p_o(t))\)  containing \(t\) , for any \(t\in U\) .
For a star-shaped open set \(U\)  with respect to \(U\cap S_o\) , the flow of the radial vector field in each stratum \(S\)  containing \(S_o\)  in its closure induces a deformation retraction of \(U\cap S\)  to \(U\cap S_o\) . On the other hand, there exists a fundamental system of open neighbourhoods \(V_S\)  of \(U\cap S_o\)  in \(U\cap S\)  which are star-shaped. So \(V_S\cap S\subset U\cap S\)  induces an isomorphism of fundamental groups.
Assume we are given \((^{t_o},_{0})\)  with model \(\gamma _{t_o}^+\)  for \(t_o\in S_o\) . Since \(_T()_{|S_o}\)  is locally constant with fibre \(_0(\gamma _{t_o}^+)\)  (Proposition REF ) and \(U\cap S_o\)  is simply-connected, \((^{t_o},_{0})\)  extends in a unique way as a section of \(_T()_{|U\cap S_o}\) , and we find \((,_{T})\)  defined in some neighbourhood of \(U\cap S_o\)  in \(T\) . Given a stratum \(S\)  with \(S\supset S_o\) , \((,_{T})\)  is defined on some \(V_S\)  as above and defines a section of \(_T()_{|V_S\cap S}\) . Since \(_T()_{|S}\)  is locally constant and since \(\pi _1(V_S\cap S,\star )\rightarrow \pi _1(U\cap S,\star )\)  is an isomorphism, this section extends to \(U\cap S\)  (Corollary REF ) and we obtain \((,_{T})\)  on \(U\cap S\) , hence on a neighbourhood of \(U\cap S\)  in \(U\) . Since \(U\)  does not cut any stratum \(S^{\prime }\)  such that \({S^{\prime }}\supsetneq S_o\) , it is covered by the strata \(U\cap S\)  with \(S\)  such that \(S\supset S_o\) . By uniqueness (Corollary REF ), the extensions on the neighbourhoods of the various strata glue together, and give rise to \((,_{T})\)  on \(U\) .

Proof of Corollaries REF  and  REF
The first part of Corollary REF  is contained in the theorem. For the second part, let \(t_o\in V\)  and let \((,_{T})_V\)  and \((^{\prime },^{\prime }_{T})_V\)  be two \(\) -marked \(T\) -meromorphic flat bundles on \(V\times (_z,0)\)  whose restriction at \(t_o\)  are equal to \((^{t_o},_{0})\) . Then, by Theorem , \((,_{T})_V\)  and \((^{\prime },^{\prime }_{T})_V\)  coincide on some \((t_o)\subset T\) , hence on a nonempty open set in \(V(\Delta \cap V)\) . Since the latter is connected, they coincide on \(V(\Delta \cap V)\) , by an argument similar to that used in Corollary REF . We conclude with Lemma REF .
[Proof of Corollary REF ]
The surjectivity of \(\gamma _{y_o}^+\)  and the injectivity of \(f^+\)  are obvious from Theorem , since we can identify \((^{y_o},_{0})\)  with \((^{t_o},_{0})\) , due to the identification \(\gamma _{y_o}^+_Y=\gamma _{t_o}^+\) , and we have \(\gamma _{t_o}^+=\gamma _{y_o}^+\circ f^+\) . It is then enough to prove the injectivity of \(\gamma _{y_o}^+\) . This is an immediate consequence of Corollary REF .

Application to the sheaf of Stokes torsors
We note that the restriction of \(\)  to any stratum of the natural stratification of \(T\)  compatible with \(\Delta \)  is good, hence for each such stratum \(S\) , the sheaf \(_S(i_S^+)\simeq _S(i_S^+)\)  is a locally constant sheaf. On the other hand, we do not have much information on the sheaf \(_T()\) , except on the open dense stratum \(T\Delta \) . However, the sheaf \(_T()\)  is better behaved, and this will enable us to compare the various sheaves \(_S(i_S^+)\) . Indeed, Theorem  can be interpreted as a constructibility theorem for the sheaf \(_T()\) .
Let us fix a stratum \(S_o\)  of \(\Delta \)  in \(T\) , let \(U\)  be an open subset of \(T\)  containing \(S_o\)  and satisfying (REF ) and (REF ), and let us consider the following diagram:
\(@C=1.5cm@R=.3cm{&S_o\times ^m&\\S_o=S_o\times \lbrace 0\rbrace @{^{ (}->}[r]^-{i_o}[ddr]_{}& U@{}[u]|\bigcup [dd]^{p_o}&@{_{ (}->}[l]_-{j_o}U\Delta [ddl]^{q_o}\\&&\\&S_o&}\) 
(the notation in the proof of Proposition REF ). We also denote by \(i_o,j_o\)  the complementary inclusions of \(S_o\)  and \(TS_o\)  in \(T\) .
[Constructibility of \(_T()\) ]
The sheaf \(_T()\)  is constructible, and more precisely, for each stratum \(S_o\)  of \(\Delta \) , each of the natural morphisms

po*T()|Uio-1T()io+So(iSo+)
is an isomorphism (the left one comes from the sheaf-theoretic adjunction \(\!\rightarrow \!i_{o*}i_o^{-1}\) ), and the natural composed morphism
\(_{S_o}(i_{S_o}^+)i_o^{-1}_T()\rightarrow i_o^{-1}j_{o*}j_o^{-1}_T()\) 
is injective. Moreover, the natural restriction morphism
\(p_{o*}_T()_{|U}\rightarrow q_{o*}_T()_{|U\Delta }\) 
is injective. Lastly, the natural morphism that one deduces from it together with (REF ) and the sheaf-theoretic adjunction \(q_o^{-1}q_{o*}\rightarrow \) :

qo-1So(io+)T()|U
is also injective. The image subsheaf of (REF ) is characterized as follows: given \(t\!\in \!U\Delta \) , setting \(t_o\!=\!q_o(t)\!\in \!S_o\) , a germ of section \((^t,_{T,t})\in _{T\Delta }()_t\simeq _0(i_t^+)\)  belongs to (the image of) \([q_o^{-1}_{S_o}(i_o^+)]_t\simeq _{S_o}(i_o^+)_{t_o}\)  if and only if there exists a neighbourhood \((t_o)\)  of \(t_o\)  in \(S_o\)  and \((,_{T})\)  on \(p_o^{-1}((t_o))\)  whose germ at \(t\)  is \((^t,_{T,t})\) .
The statement is local on \(S_o\) . Each \(t_o\in S_o\)  has a fundamental system of 1-connected neighbourhoods \(V\)  in \(S_o\) . On the other hand, each such \(V\)  has a fundamental system of neighbourhoods \(U^{\prime }\)  in \(T\)  satisfying the properties (REF )–(REF ). Let us denote by \(i_o\)  the inclusion \(S_oT\) . We wish to prove that for each such pair \((V,U^{\prime })\) , the restriction map \(i_o^+:\Gamma (U^{\prime },_T())\rightarrow \Gamma (V,_{S_o}(i_o^+))\)  is a bijection, as the case \(U^{\prime }=p_o^{-1}(V)\)  also implies that the composed morphism (REF ) is an isomorphism.
Since \(V\)  is simply connected and \(_{S_o}(i_o^+)\)  is locally constant (Proposition REF ), we have \(\Gamma (V,_{S_o}(i_o^+))=i_{t_o}^{-1}_{S_o}(i_o^+)\) . Moreover, the same proposition identifies \(i_{t_o}^{-1}_{S_o}(i_o^+)\)  with \(_0(i_{t_o}^+)\)  via \(\gamma _{t_o}^+\) . Lastly, Theorem  implies that \(\gamma _{t^0}^+:\Gamma (U^{\prime },_T(N))\rightarrow _0(i_{t_o}^+)\)  is a bijection. We conclude that \(\Gamma (U^{\prime },_T())\rightarrow \Gamma (V,_{S_o}(i_o^+))\)  is also a bijection. We then use Corollary REF  to conclude the first assertion.
For the second assertion, we need to prove that if \((,_{T})\)  and \((^{\prime },^{\prime }_{T})\)  defined on \(p_o^{-1}(V)\)  (\(V\)  as above) coincide on \(q_o^{-1}(V)\) , then they coincide. This follows from Hartogs theorem (Lemma REF ).
For the (REF ), we are left to proving that the adjunction morphism
\(q_o^{-1}q_{o*}_T()_{|U\Delta }\rightarrow _T()_{|U\Delta }\) 
is injective. This amounts to proving that, if two sections of \(_T()\)  on \(q_o^{-1}(V)\)  (\(V,t_o\)  as above) coincide at \(t\in q_o^{-1}(t_o)\) , they coincide everywhere on \(q_o^{-1}(V)\) . Since \(q_o^{-1}(V)\)  is also connected by the star-shaped property of \(U\) , we can apply the same argument as in Corollary REF  to obtain the desired property.
Let us prove the last assertion of the corollary. That \((^t,_{T,t})\in _{T\Delta }()_t\)  belongs to the image of \([q_o^{-1}_{S_o}(i_o^+)]_t\)  means that there exists \((^{t_o},_{T,t_o})\in _{S_o}(i_o^+)_{t_o}\) , that we consider as a section \(\sigma _o:S_o\supset (t_o)\rightarrow _{S_o}(i_o^+)^\)  of the sheaf (étalé) space \(_{S_o}(i_o^+)^\rightarrow S_o\)  (which is a covering since \(_{S_o}(i_o^+)\)  is locally constant) such that, if \(\sigma :(t)\rightarrow _{T\Delta }()^\)  denotes the section corresponding to \((^t,_{T,t})\) , we have \(\sigma (t)=(\sigma _o\circ q_o)(t)\) , regarding \(\sigma _o\circ q_o\)  as a section
\(q_o^{-1}((t_o))\rightarrow [q_o^{-1}_{S_o}(i_o^+)]^\rightarrow _{T\Delta }()^\) 
of the of the sheaf space \(_{T\Delta }()^\)  over \(T\Delta \)  (which is also a covering). By definition , \(\sigma _o\circ q_o\)  corresponds to the restriction to \(q_o^{-1}((t_o))\)  of \((,_{T})\)  constructed by Theorem  on \(p_o^{-1}((t_o))\)  from \((^{t_o},_{0})\)  corresponding to \(\sigma _o\) . Then the germ \((,_{T})_t\)  corresponds to \(\sigma \)  by the correspondence (REF ).
The converse is obtained similarly.
On the one hand, let us consider the sheaf \(_{S_o}(i_o^+)=_{S_o}(i_o^+_o)\)  ((REF )). Since \(_o\)  is good on \(S_o\) , this is a locally constant sheaf of pointed sets on \(S_o\) . On the other hand, we have the locally constant sheaf \(_{T\Delta }()\)  since \(\)  is good on \(T\Delta \) .
[Comparison of sheaves of Stokes torsors]
There exists natural injective morphisms

So(io+)io-1jo*T().

and

qo-1So(io+)T()|U.
The image subsheaf of (REF ) is characterized as follows: given \(t\!\in \!U\Delta \) , setting \(t_o\!=\!q_o(t)\!\in \!S_o\) , a germ of section \(\sigma _t\in _{T\Delta }()_t\simeq _0(i_t^+)\)  belongs to (the image of) \([q_o^{-1}_{S_o}(i_o^+)]_t\simeq _{S_o}(i_o^+)_{t_o}\)  if and only if there exists a neighbourhood \((t_o)\)  of \(t_o\)  in \(S_o\)  and \((,_{T})\)  on \(p_o^{-1}((t_o))\)  whose germ at \(t\)  corresponds to \(\sigma _t\)  via (REF ).
The morphism (REF ) is defined by the following diagram:
\(@C=1.5cm{_{S_o}(i_o^+)[d]_\wr @{-->}[r]&i_o^{-1}j_{o*}_{T\Delta }()[d]^\wr \\i_o^{-1}_T()@{^{ (}->}[r]&i_o^{-1}j_{o*}j_o^{-1}_T()}\) 
The left vertical isomorphism is obtained by composing the isomorphism of Theorem REF  on \(S_o\)  (\(_{S_o}(i_o^+)\simeq _{S_o}(i_o^+)\) ) together with the isomorphism of Corollary REF . The right vertical isomorphism is given by Theorem REF  on \(T\Delta \) . Lastly, the lower horizontal morphism is injective, according to Corollary REF . We define (REF ) similarly and use (REF ) for its injectivity. The last assertion is obtained similarly from the last assertion in Corollary REF .

Isomonodromic deformations


For a given square matrix \(M\) , we denote by \(M=M^+M^\)  the decomposition into the diagonal and non-diagonal parts.

For \(t_o\in \Delta \) , we decompose \(\lbrace 1,\dots ,n\rbrace =\bigsqcup _{r\in R}I_r\)  such that, for every \(r\in R\) , we have \(\lbrace i,j\rbrace \subset I_r\)  if and only if \(t_{o,i}=t_{o,j}\) .

Universal deformation of a Birkhoff normal form
For \(t\in T\) , we denote by \(\Lambda (t)\)  the matrix \((t_1,\dots ,t_n)\) . For \(t_o\in T\) , say that a system with matrix \(({\Lambda (t_o)}{z}+B(z)){z}{z}\) , with \(B(z)\)  holomorphic, is in the Birkhoff normal form if \(B(z)=A_1\)  is constant, that is,
\(\Bigl (\frac{\Lambda (t_o)}{z}+A_1\Bigr )\frac{z}{z}.\) 
For the system (REF ), if \(t_o\notin \Delta \) , a theorem of B. Malgrange [8]}, [9]} asserts that there exists a universal integrable deformation of this system in the neighbourhood of \(t_o\)  (also [10]}). In particular ([10]}), there exists a holomorphic matrix \(F^_1(t)\)  near \(t_o\)  with zeros on the diagonal, such that the system of the form (REF )
\(\Bigl (\frac{\Lambda (t)}{z}+[\Lambda (t),F^_1(t)]+A^_1]\Bigr )\frac{z}{z},\) 
where \(A^_1\)  is the diagonal part of \(A_1\) , is integrable and restricts to (REF ) at \(t=t_o\) . The integrable connection (on the trivial bundle) has matrix ([10]})
\(-(\Lambda (t)/z)+\bigl ([\Lambda (t),F_1^(t)]+A_1^\bigr )\,\frac{z}{z}-[\Lambda (t),F_1^(t)]\) 
and is a universal integrable deformation of its restriction at each point of the neighbourhood where it exists. Moreover, there exists a \(z\) -formal base change which transforms (REF ) to the system
\(-(\Lambda (t)/z)+A_1^\,\frac{z}{z}.\) 
The following two questions are natural:

(REF )
If \(t_o\in \Delta \) , can we find an integrable deformation (REF ) of the Birkhoff normal form (REF ) with \(z\) -formal normal form (REF )?

(REF )
Given a system (REF ) defined on an open set \(V\)  of \(T\)  (so that its restriction at every \(t\in V\)  is in the Birkhoff normal form), and given a matrix \(F_1^(t)\) , holomorphic on \(V\) , assume that, in a small neighbourhood \(W\)  of \(t_o\in V\Delta \) , (REF ) underlies a universal integrable deformation (REF ) of its restriction at \(t=t_o\) . Does this integrable deformation extend on \(V\) , in particular at points \(t\in \Delta \cap V\) , and has \(z\) -formal normal form (REF ) on \(V\) ?
As an application of Theorem  we give an answer to (REF ) in Section REF . On the other hand, the results of [13]} and [14]} concern (REF ).

Deformation of a Birkhoff normal form
We will show that Question (REF ) for (REF ) has a positive answer provided that Assumptions REF (REF ) and (REF ) below are fulfilled by \(A_1\) . We moreover make more precise the domain of existence of the integrable deformation.
Let \(t_o\in \Delta \subset T\)  and let us fix a Birkhoff normal form (REF ).
With the following assumptions, which are always fulfilled if \(t_o\notin \Delta \) :

\(A_1^\in (\Lambda (t_o))\) ,Recall that \((\Lambda (t_o))(B)=[\Lambda (t_o),B]\)  so that \(A\in (\Lambda (t_o))\)  iff \(A_{ij}=0\)  whenever \(i,j\in I_r\)  for some \(r\in R\) .


\(A_1^\)  is partially non-resonant,
\(\forall r\in R,\;\forall i,j\in I_r,\quad A_1^{}_{ii}-A_1^{}_{jj}\notin \lbrace 0\rbrace ,\) 
there exists a \(z\) -formal base change of the form \(+zP_1+\cdots \)  which transforms the matrix of \(\nabla ^o\)
\(A(t_o,z)=\Big (\frac{\Lambda (t_o)}{z}+A_1\Big )\frac{z}{z}\) 
to the diagonal matrix
\(A^(t_o,z)=\Big (\frac{\Lambda (t_o)}{z}+A_1^\Big )\frac{z}{z}.\) 
The result is standard ([13]}), but we will give a proof valid on any algebraic closed field of characteristic zero instead of \(\) , not depending on the transcendental notion of fundamental solution. Firstly, on can find a formal base change \(F_o=-zF_1^(t_o)+\cdots \)  which transforms \(A(t_o,z)\)  to the formal matrix
\(A(t_o,z)=\Big (\frac{\Lambda (t_o)}{z}+A_1^+zA_2+\cdots \Big )\frac{z}{z},\) 
where \(A_2,\dots \)  are block-diagonal with respect to \(r\in R\) , so that \(A(t_o,z)\)  is also block-diagonal. Each block (\(r\in R\) ) is written as
\(A^{(r)}(t_o,z)=\Big (\frac{t_{o,r}^{(r)}}{z}+A_1^{(r)}+zA^{(r)}_2+\cdots \Big )\frac{z}{z}.\) 
Since \(A_1^{(r)}\)  is non-resonant, there exists a base change \(^{(r)}+zQ^{(r)}_1+\cdots \)  which transforms \(A^{(r)}(t_o,z)\)  to the \(r\) th block of \(A^(t_o,z)\) .
Let \(t_o\in \Delta \)  and let \(U\)  be a connected open subset of \(T\)  satisfying (REF )–(REF ) with respect to \(S(t_o)\) . Under Assumptions REF (REF ) and (REF ) on \(A_1\) , there exists a holomorphic hypersurface \(\Theta \)  in \(U\)  not containing \(t_o\)  and a holomorphic matrix \(F_1^(t)\)  on \(U\Theta \) , such that the meromorphic connection on the trivial bundle with matrix (REF ) is integrable, restricts to (REF ) at \(t_o\) , and is formally equivalent at \(z=0\)  to the matrix connection
\(-(\Lambda (t)/z)+A_1^\,\frac{z}{z}.\) 
The proof is very similar to that of [8]}, [9]}. We set \(T=\lbrace z=0\rbrace \subset X=^n\times _z\) . We denote by \(\)  the \(T\) -meromorphic flat bundle \(\bigoplus _{i=1}^n(^{-t_i/z}\otimes _i)\) , where \(_i\!=\!(_X(*T),+A_1^{}_{ii}\,z/z)\) . By Lemma REF , (REF ) defines an object \((^{t_o},_{0})\)  in \(_0(\gamma _{t_o}^+)\) . From Theorem  we deduce an \(\) -marked \(T\) -meromorphic flat bundle \((,_{T})\)  on \(U\times (_z,0)\) . We can now apply [10]} and obtain a hypersurface \(\Theta \subset U\)  and a basis \(\)  of \((*(\Theta \times _z))\)  in which the matrix of the integrable connection takes the form
\(\Big (\frac{A_0(t)}{z}+A_1\Big )\frac{z}{z}+\frac{C(t)}{z}\) 
with \(A_0(t)\)  conjugate to \(\Lambda (t)\)  for each \(t\in U\Theta \)  and \(C(t)\in \Gamma (U,\Omega ^1_T(*\Theta ))\) . One can then apply the results in [10]}, since the regularity property of \(\Lambda (t)\)  (the fact that the eigenvalues are pairwise distinct) is not needed at this point. The change of notation with respect to  is as follows: \(\Delta _0\)  is \(\Lambda (t)\) , \(\Delta _\infty \)  is \(A_1^\) , \(\tau \)  is \(z\)  and the non-diagonal part \(T^\)  is \(F_1^(t)\) .
Application to the construction of Frobenius manifolds
It is known ([20]}) that, given suitable initial data consisting of a diagonal matrix \(\Lambda (t_o)\)  with pairwise distinct eigenvalues (\(t_o\notin \Delta \) ), of a matrix \(A_1\)  such that \(A_1-(w/2)\)  is skew-symmetric for some integer \(w\) , and an eigenvector \(\omega _o\)  of \(A_1\)  which has no zero entry, one can construct a Frobenius manifold structure on the complement of some hypersurface \(\Theta \)  in the universal covering \({T\Delta }\)  (also [10]}). It is the universal model at a semisimple point of a Frobenius manifold.
Theorem REF  enables us to relax the regularity assumption on \(\Lambda (t_o)\) , provided that Assumptions REF (REF ) and (REF ) are fulfilled for \(A_1\) . Let \(t_o\in \Delta \)  and set \(S_o=S(t_o)\) . Assume that \((\Lambda (t_o),A_1,\omega _o)\)  satisfy the following properties.

\(A_1^\in (\Lambda (t_o))\) ,


\(A_1-(w/2)\)  is skew-symmetric (so that \(A_1^=(w/2)\)  is non-resonant),


\(\omega _o\)  is an eigenvector of \(A_1\)  whose entries are all nonzero.

Let \(U\)  be an open subset of \(T\)  containing \(S_o\)  and satisfying (REF ) and (REF ), and let \(U\)  be its universal covering. By Theorem REF , there exists an integrable deformation of the Birkhoff normal form (REF ), which exists on \(U\Theta \)  for some complex hypersurface \(\Theta \)  of \(U\) . The vector \(\omega _o\)  can be extended flatly as a vector function \(\omega \)  on \(U\) , meromorphic along \(\Theta \)  and its entries do not vanish away from some hypersurface \(\Theta _{\omega _o}\) . To \(\omega \)  is then associated an infinitesimal period mapping ([10]}).
Under Assumptions (REF ), (REF ) and (REF ) above, the infinitesimal period mapping associated with \(\omega \)  endows the manifold \(U(\Theta \cup \Theta _{\omega _o})\)  with a Frobenius structure.
